An eligible \(W\)  can be constructed by the Metropolis rule [1]}, [2]}. assum2 implies that \(-1<\lambda _n(W)\le \lambda _2(W)<1\) . We define the spectral gap \(\rho =1-\lambda _2(W)\in [0,1)\)  to characterize the effect of \(W\)  on the convergence rate. Under assum2, Problem original is equivalent to the following optimization problem
\(\begin{alignedat}{-1}&\operatornamewithlimits{minimize}_{\mathbf {x}\in \mathbb {R}^d}\ F(X)\triangleq \sum _{i=1}^n f_i(\mathbf {x}_i),\\&\operatornamewithlimits{subject\ to}\ (I-W)X=0.\end{alignedat}\) 
